The integration algorithm of Lax equation for both generic Lax matrices and generic initial conditions

Several physical applications of Lax equation require its general solution for generic Lax matrices and generic not necessarily diagonalizable initial conditions. In the present paper we complete the analysis started in [arXiv:0903.3771] on the integration of Lax equations with both generic Lax operators and generic initial conditions. We present a complete general integration formula holding true for any (diagonalizable or non-diagonalizable) initial Lax matrix and give an original rigorous mathematical proof of its validity relying on no previously published results.